Polynomial Times Monomial

Clearly 2(5 + 3) equals 2 \cdot 5 + 2 \cdot 3, since both are 16. In the same way a(b + c) = ab + ac. This is the Distributive Law of Multiplication.

Rule. Multiply each term of the polynomial by the monomial, and write the results one after another with their proper signs.

\begin{array}{r} 3x^2 - 2xy + 4y - 5a - 6 \\ 2xy \\ \hline 6x^3y - 4x^2y^2 + 8xy^2 - 10axy - 12xy \end{array}

These laws are obvious for whole numbers, but they stay true when letters stand for negative numbers, fractions, algebraic expressions, and kinds of numbers we'll meet later. The idea that every kind of number follows the commutative, associative, and distributive laws is often called the Law of Permanence.